Frey curve

In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve

${\displaystyle y^{2}=x(x-\alpha )(x+\beta )}$

associated with an ABC triple ${\displaystyle \alpha +\beta =\gamma }$. This relates properties of solutions of equations to elliptic curves. This curve was popularized in its application to Fermat’s Last Theorem where one investigates a (hypothetical) solution of Fermat's equation

${\displaystyle a^{\ell }+b^{\ell }=c^{\ell }.}$

The curve is named after Gerhard Frey and (sometimes) Yves Hellegouarch.